Srinivasa Ramanujan:Going Strong at 125,Part IKrishnaswami Alladi, Editor

The 125th anniversary of the birth of the Indianmathematical genius Srinivasa Ramanujan fallson December 22, 2012. To mark this occasion, theNotices offers the following feature article, whichappears in two installments. The article contains anintroductory piece describing various major devel-opments in the world of Ramanujan since his centen-nial in 1987, followed by seven pieces describing im-portant research advances in several areas of math-ematics influenced by him. The first installment inthe present issue of the Notices contains the intro-ductory piece by Krishnaswami Alladi, plus piecesby George Andrews, Bruce Berndt, and JonathanBorwein. Pieces by Ken Ono, K. Soundararajan, R. C.Vaughan, and S. Ole Warnaar will appear in thesecond installment in the January 2013 issue of theNotices.

Krishnaswami Alladi

Ramanujan’s Thriving Legacy

Srinivasa Ramanujan is one of the greatest mathe-maticians in history and one of the most romanticfigures in the mathematical world as well. Bornon December 22, 1887, to a poor Hindu brahminfamily in Erode in the state of Tamil Nadu inSouth India, Ramanujan was a self-taught geniuswho discovered a variety of bewildering identitiesstarting from his school days. There is a legend thatthe Hindu goddess Namagiri in the neighboringtown of Namakkal used to come in his dreams andgive him these formulae. Unable to find individualsin India to understand and evaluate his findings,Ramanujan wrote two letters in 1913 to G. H.Hardy of Cambridge University, England, listingseveral of his most appealing formulae. The depth

Krishnaswami Alladi is professor of mathematics at the Uni-versity of Florida, Gainesville. His email address is alladik@ufl.edu.

DOI: http://dx.doi.org/10.1090/noti917

and startling beauty of these identities convincedHardy that Ramanujan was on a par with Euler andJacobi in sheer manipulative ability. At Hardy’sinvitation, Ramanujan went to Cambridge in 1914.The rest is history. In the five years he was inEngland, Ramanujan published several importantpapers, some of which stemmed from his discover-ies in India; he collaborated with G. H. Hardy andwrote two very influential papers with him. Forhis pathbreaking contributions, Ramanujan washonored by being elected Fellow of Trinity College,Cambridge, and Fellow of the Royal Society (FRS)even though he did not have a college degree! Therigors of life in England during the First World War,combined with his own peculiar habits, led to arapid decline in his health. Ramanujan returned toMadras, India, in 1919 a very sick man and diedshortly after on April 20, 1920. Even during hislast days, his mathematical creativity remainedundiminished. He wrote one last letter to Hardyin January 1920 outlining his discovery of themock theta functions, considered to be amonghis deepest contributions. In the decades afterRamanujan’s death, we have come to realize thedepth, breadth, and significance of his many discov-eries. Ramanujan’s work has had a major influenceon several branches of mathematics, most notablyin number theory and classical analysis, and evenin some areas of physics.

During the Ramanujan Centennial in 1987,eminent mathematicians from around the worldgathered in India to pay homage to this singulargenius. It was an appropriate time to reflect on hislegacy and consider the directions in which his workwould have influence in the future. In this quartercentury since the centennial, many significantdevelopments have taken place in the world ofRamanujan: (i) there have been major researchadvances in recent years that have widened thearena of impact of Ramanujan’s work; (ii) bookson Ramanujan’s work, of interest to students

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and researchers alike, and books on Ramanujan’slife with appeal to the general public have beenpublished; (iii) journals bearing Ramanujan’s namehave been established, one of which is devotedto all areas of research influenced by him; (iv)annual conferences devoted to various aspects ofRamanujan’s work are being held in his hometownin India; (v) international prizes bearing his nameand encouraging young mathematicians have beencreated; and (vi) movies and plays about Ramanujaninspired by his remarkable life have also appearedor are in production.

This feature in the Notices of the AMS is devotedto describing some of the significant developmentsin the world of Ramanujan since the centennial.My opening article will broadly address the devel-opments listed under (i)–(vi). This will be followedby articles by leading mathematicians on variousaspects of research related to Ramanujan’s work.

Some Major Research AdvancesI focus here on certain aspects of the theory ofpartitions and q-hypergeometric series with whichI am most familiar and provide just a sampleof some very important recent results. Researchrelating to Ramanujan’s work in many other areashas been very significant, and some of these ideaswill be described by others in the following articles.

Ramanujan’s mock theta functions are mysteri-ous, and the identities he communicated to Hardyabout them are deep. Since George Andrews un-earthed the Lost Notebook of Ramanujan at TrinityCollege Library in Cambridge University in 1976,he has been analyzing the identities containedtherein, especially those on the mock theta func-tions. We owe much to Andrews for explainingthe mock theta function identities in the contextof the theory partitions (see for instance [12]). InAndrews-Garvan [21] the mock theta conjecturesare formulated. These relate the q-hypergeometricmock theta function identities to partition identi-ties; the mock theta conjectures were proved byHickerson [40] around the time of the centennial.

In spite of their resemblance to the classicaltheta functions, the exact relationship between themock theta functions and theta functions remainedunclear. In the last decade, dramatic progress onthis matter has been achieved by Ken Ono andhis coworkers, most notably Kathrin Bringmann,in establishing the links between mock thetafunctions and Maass forms, thereby providing thekey to unlock this mystery [32], [33], [34].

Ramanujan’s congruences for the partition func-tion modulo the primes 5, 7, and 11 and theirpowers are among his most significant discov-eries. More specifically, Ramanujan discoveredthat p(5n + 4) is a multiple of 5, p(7n + 5) isa multiple of 7, and p(11n + 6) is a multiple of

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The bust of Ramanujan now stands in the centralhall of his home in Kumbakonam. The bust wasadded after SASTRA University purchasedRamanujan’s home. In the background is areplica of his famous passport photograph.

11, where p(n) is the number of partitions ofn. In the ensuing decades, this led to a fruitfulstudy of congruences satisfied by coefficients ofmodular forms. Yet it was unclear whether thepartition function itself satisfied congruences withrespect to other primes. Oliver Atkin [22] wasthe first to discover congruences for the partitionfunction of the type p(An+ b) ≡ C(mod p) with aprime modulus p > 11, but unlike the Ramanujancongruences, we have A not equal to p. In recentyears Scott Ahlgren [1] and Ken Ono [44] havediscovered partition congruences to larger primemoduli and have shown that there are indeedinfinitely many such congruences.

Ramanujan (see [46]) wrote down two spec-tacular identities for the generating functions ofp(5n+ 4) and p(7n+ 5), from which his congru-ences modulo 5 and 7 follow. Proofs of Ramanujan’spartition congruences were given by Atkin usingthe theory of modular forms. In 1944, FreemanDyson [35] gave a combinatorial explanation ofRamanujan’s partition congruences mod 5 and 7using the now famous statistic he called the rank.He then conjectured the existence of a statistic

December 2012 Notices of the AMS 1523

he dubbed the crank that would explain the mod11 congruence. A combinatorial explanation ofRamanujan’s partition congruence modulo 11 wasfirst given in the Ph.D. thesis of Frank Garvan [36]using vector partitions and a vector crank. This wasthen converted to ordinary partitions by Andrewsand Garvan [20] when they got together at the Ra-manujan Centennial Conference at the Universityof Illinois, Urbana, in the summer of 1987, and sofinally the crank conjectured by Dyson was found.Perhaps the Goddess of Namakkal made surethat the solution to this problem should actuallyhappen during the Ramanujan Centennial! Since1987 there has been an enormous amount of workon the rank and on cranks for various partition sta-tistics by several mathematicians: Garvan, Kim, andStanton [37] found cranks for t-cores; Mahlburg[43] found cranks for the partition congruencesto moduli larger than 11; and Andrews [16] hasnoticed new connections with Durfee symbols ofpartitions. In the last decade, Richard Stanley [49]introduced a refinement of Dyson’s rank, and thishas led to important refinements and extensionsof Ramanujan’s partition congruences (see forinstance Andrews [15] and Berkovich-Garvan [24]).

In the entire theory of partitions andq-hypergeometric series, the Rogers-Ramanujanidentities are unmatched in simplicity of form, ele-gance, and depth. This pair of identities connectspartitions into parts that differ by at least twowith partitions into parts congruent to ±i(mod 5),for i = 1,2. Nowadays, by a Rogers-Ramanujan(R-R) type identity, we mean an identity whichin its analytic form equates a q-hypergeometricseries to a product, where the series representsthe generating function of partitions whose partssatisfy gap conditions and the product is thegenerating function of partitions whose partssatisfy congruence conditions. This subject hasblossomed into an active area of research thanksto systematic analysis of R-R type identities,primarily by Andrews ([11], Chapter 7). In the1980s the Australian physicist Rodney Baxtershowed how certain R-R type identities arise assolutions of some models in statistical mechanics(see Andrews [13], Chapter 8). Soon after, Andrews,Baxter, and Forrester [18] determined the classof R-R type identities that arise in such modelsin statistical mechanics. For this work Baxterwas recognized with the Boltzmann Medal of theAmerican Statistical Society.

Subsequently there has been significant work onq-hypergeometric identities and integrable modelsin conformal field theory in physics. The leaderin this effort is Barry McCoy, who in the 1990s,with his collaborators Alexander Berkovich, AnneSchilling, and Ole Warnaar [25], [26], [27], [28],discovered new R-R type identities from the study

of integrable models in conformal field theory.For this work McCoy was recognized with theHeineman Prize in Mathematical Physics and aninvited talk at the ICM in Berlin in 1998.

My own work in the theory of partitions andq-hypergeometric series has been in the studyof R-R type identities and their refinements [4],especially on a partition theorem of Göllnitz, oneof the deepest R-R type identities. In collaborationwith Basil Gordon, George Andrews, and AlexanderBerkovich, I was able to obtain generalizationsand refinements of Göllnitz’s partition theoremin 1995 [9] and later, in 2003, a much deeperfour-parameter extension of the Göllnitz theorem[10].

Books on Ramanujan’s Life and WorkHardy’s twelve lectures on Ramanujan [39] area classic in exposition. For several decades afterRamanujan’s death, this book and the CollectedPapers [46] of Ramanujan were the primary sourcesfor students and young researchers aspiring toenter the Ramanujan mathematical garden. In 1987,The Lost Notebook of Ramanujan was published[48]. Students today are very fortunate, becauseBruce Berndt has edited the original three note-books of Ramanujan and published this materialin five volumes [29]. In these volumes one can findproofs of Ramanujan’s theorems and how they arerelated to past and contemporary research. For thismonumental contribution, Berndt was recognizedwith the AMS Steele Prize. In the past five years,Bruce Berndt and George Andrews have joinedforces to edit Ramanujan’s Lost Notebook. This isexpected to also require five volumes, of whichthree have already appeared [19].

Ramanujan is now making an impact beyondmathematics into society in general around theworld. Throughout India, Ramanujan’s life storyis well known, and Ramanujan is a hero to everyIndian student. But with the publication of RobertKanigel’s book The Man Who Knew Infinity [41],Ramanujan’s life story has reached the world over.The impact of this book cannot be underestimated(see [2]). Subsequently, Bruce Berndt and RobertRankin have published two wonderful books. Thefirst one, called Ramanujan—Letters and Com-mentary [30], collects various letters written to,from, and about Ramanujan and makes detailedcommentaries on each letter. For instance, if aletter contains a mathematical statement, thereis an explanation of the mathematics with appro-priate references. If there is a statement aboutRamanujan being elected Fellow of the Royal So-ciety, there is a description about the proceduresand practices for such an election (see review [3]).The second book, Ramanujan—Essays and Surveys[31], is a collection of excellent articles by various

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experts on Ramanujan’s life and work and aboutvarious individuals who played a major role inRamanujan’s life (see review [5]). This book alsocontains an article by the great mathematicianAtle Selberg, who describes how he was inspiredby Ramanujan’s mathematical discoveries. Mostrecently, David Leavitt has published a book [42]entitled The Indian Clerk which focuses on theclose relationship between Ramanujan and Hardy(see review [38]). All of these books will not onlyappeal to mathematicians but to students and laypersons as well.

Since the Ramanujan Centennial, I have writtenarticles annually comparing Ramanujan’s life andmathematics with those of several great mathe-maticians in history, such as Euler, Jacobi, Galois,Abel, and others. These articles, which were writ-ten to appeal to the general public, appeared inThe Hindu, India’s national newspaper, each yeararound Ramanujan’s birthday in December. Thecollection of all these articles, along with my bookreviews and other Ramanujan-related articles, hasjust appeared as a book [8] for Ramanujan’s 125thanniversary.

There are many other books on Ramanujanthat have been published, and my list above iscertainly not exhaustive. However, I must add tothis list the comprehensive CD on Ramanujanthat Srinivasa Rao brought out in India a fewyears ago. In this CD we have scanned imagesof Ramanujan’s notebooks, his papers, severalarticles about Ramanujan, many important letters,etc.

Journals Bearing Ramanujan’s NameInspired by what I saw and heard during the Ra-manujan Centennial, I wanted to create somethingthat would be a permanent and continuing memo-rial to Ramanujan, namely, The Ramanujan Journal,devoted to all areas of mathematics influencedby Ramanujan. This idea received enthusiasticsupport from mathematicians worldwide, many ofwhom have served, or are serving, on the editorialboard. The journal was launched in 1997 by KluwerAcademic Publishers, which published four issuesof one hundred pages per year in one volume.Now the journal is published by Springer, whichpublishes nine issues per year of one hundred fiftypages each in three volumes. The rapid growth ofthis journal is a further testimony to the fact thatRamanujan’s work is continuing to have majorinfluence on various branches of mathematics.

There is another journal bearing Ramanujan’sname: The Journal of the Ramanujan MathematicalSociety, which is distributed by the AMS and isdevoted to all areas of mathematics. It bears Ra-manujan’s name since he is India’s most illustriousmathematician.

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Left to right: Krishnaswami Alladi, GeorgeAndrews (Pennsylvania State University), andSASTRA Vice-chancellor R. Sethuraman in frontof a bust of Srinivasa Ramanujan at SASTRAUniversity, Kumbakonam—December 22, 2003.

Prior to these two journals, in the 1970s Pro-fessor K. Ramachandra (now deceased) foundedThe Hardy-Ramanujan Journal, devoted to elemen-tary and classical analytic number theory, andpublished it privately in India.

Annual Conferences in Ramanujan’sHometownA recent major event in the world of Ramanujan isthe acquisition in 2003 of Ramanujan’s home inKumbakonam, South India, by SASTRA University.This private university, founded about twentyyears ago, has grown by leaps and bounds. Sincethe university has purchased Ramanujan’s home,we have the active involvement of administrators,academicians, and students in the preservation ofRamanujan’s legacy. To mark the occasion of thepurchase of Ramanujan’s home, SASTRA Univer-sity conducted an international conference at itsKumbakonam campus December 20–22, 2003, andhad it inaugurated by Dr. Abdul Kalam, presidentof India. I was invited to bring a team of math-ematicians to this conference. George Andrewsgave the opening lecture of this conference as wellas the concluding Ramanujan CommemorationLecture on December 22, Ramanujan’s birthday. Atthe conclusion of the conference, the participantssuggested that SASTRA should conduct confer-ences annually in Kumbakonam on various areasof mathematics influenced by Ramanujan. I havehelped SASTRA organize these annual conferences.Each year several leading researchers visit SASTRAin December to speak at these conferences, andby participating in them, I have had the pleasureof spending Ramanujan’s birthday, December 22,every year in Ramanujan’s hometown (see [6]).

December 2012 Notices of the AMS 1525

Prizes Bearing Ramanujan’s NameAt the inauguration of the second Ramanujanconference at SASTRA University in December2004, Vice Chancellor Professor R. Sethuramangraciously offered to set apart US$10,000 annuallyto recognize pathbreaking research by youngmathematicians.

In response to his offer of such support, Isuggested that an annual prize be launched the nextyear to be given to mathematicians not exceedingthe age of thirty-two for outstanding contributionsto areas influenced by Ramanujan. The age limitwas set at thirty-two because Ramanujan achievedso much in his brief life of thirty-two years. Thusthe SASTRA Ramanujan Prize was born. In 2005, theyear the prize was launched, the prize committeefelt that two brilliant mathematicians, ManjulBhargava (Princeton) and Kannan Soundararajan(then at Michigan, now at Stanford), both deservedthe prize for spectacular contributions to algebraicand analytic number theory respectively. Thevice chancellor generously informed the prizecommittee that there would be two full prizesthat year, and it would not be split. The prestigeof any prize is determined by the caliber of thewinners. The SASTRA Ramanujan Prize could nothave had a better start (see [7], [45]). The prize isnow one of the most prestigious and coveted. Thesubsequent winners are: Terence Tao (UCLA) in2006, Ben Green (Cambridge University) in 2007,Akshay Venkatesh (Stanford) in 2008, KathrinBringmann (Cologne) in 2009, Wei Zhang (Harvard)in 2010, Roman Holowinsky (Ohio State) in 2011,and Zhiwei Yun (MIT and Stanford) in 2012. Thisprize has been given in December each year inRamanujan’s hometown, Kumbakonam, during theSASTRA Ramanujan Conference.

There is another very prestigious prize withRamanujan’s name attached, namely, the ICTPRamanujan Prize. The International Centre forTheoretical Physics (ICTP) in Trieste, Italy, wascreated in the sixties by the great physicist AbdusSalam with a particular commitment to encour-age scientists from developing countries. Keepingthis vision of Salam, the ICTP Ramanujan Prizelaunched in 2003 is a US$10,000 annual prize formathematicians under the age of forty-five fromdeveloping countries for outstanding research inany branch of mathematics. Whereas the SASTRARamanujan Prize is open to mathematicians theworld over but focuses on areas influenced byRamanujan, the ICTP Ramanujan Prize is open toall areas of mathematics but focuses on candidatesfrom developing countries. The ICTP Prize hasRamanujan’s name because Ramanujan is the great-est mathematician to emerge from a developingcountry. The ICTP Ramanujan Prize is given incooperation with the Abel Foundation.

Finally, there is the Srinivasa Ramanujan Medalgiven by the Indian National Science Academyfor outstanding contributions to mathematicalsciences. There is no age limit for this medal.Since its launch in 1962, the medal has been givenfourteen times.

Movies and Plays about RamanujanRamanujan’s life story is so awe inspiring thatmovies and plays about him have been and are beingproduced. The first was a superb documentaryabout Ramanujan in the famous Nova series ofthe Public Broadcasting System (PBS) on television,which described some of his most appealingmathematical contributions in lay terms and someof the most startling aspects of his life, such asthe episode of the taxi cab number 1729. Thisdocumentary was produced before the RamanujanCentennial but after Andrews’s discovery of theLost Notebook.

Since the Ramanujan Centennial, there havebeen many stage productions, including the OperaRamanujan, which was performed in Munich in1998. In 2005 I had the pleasure of seeing theplay Partition at the Aurora Theater in Berkeleyalong with George Andrews. In this and otherstage productions, there is usually a “playwright’sfancy”, namely, a modification of the true lifestory of Ramanujan for a special effect. Theproducer of Partition, Ira Hauptman, focuses onRamanujan’s monumental work with Hardy onthe asymptotic formula for the partition function,but for a special effect describes Ramanujan’sattempts to solve Fermat’s Last Theorem andRamanujan’s conversation with the Goddess ofNamakkal regarding this matter. As far as weknow, Ramanujan never worked on Fermat’s LastTheorem, but this digression for a special effect isacceptable artist’s fancy (see [8], article 23).

In 2006 Andrews and I served on a panel tocritique a script entitled A First Class Man whichhad received support from the Sloan Foundation.The reading of the script was done at the TribecaFilm Institute in New York when the distinguishedwriter David Freeman was present. Andrews andI helped in clearing up certain mathematicalmisconceptions in the script and also noted thatthe script had changed the Ramanujan life story incertain ways that were not appropriate. Regardlessof how the life of Ramanujan is portrayed in thesestage productions, they have had the positive effectof drawing the attention of the public around theworld to the life of this unique figure in history.

In 2007 a play entitled A Disappearing Num-ber was conceived and directed by the Englishplaywright Simon McBurney for the Theatre Com-plicite Company. It first played at the TheaterRoyal in Plymouth, England, and won three very

1526 Notices of the AMS Volume 59, Number 11

prestigious awards in England in 2007. This playwas performed at the International Congress ofMathematicians in Hyderabad, India, in August2010.

The latest theatrical production is a movie thatis now being produced in India based on Kanigel’sbook The Man Who Knew Infinity and featuring oneof the most popular actors in the Tamil cinemaworld as Ramanujan.

Events in 2011–12Since December 2012 is the 125th anniversary ofRamanujan’s birth, the entire year starting fromDecember 22, 2011, was declared as the Year ofMathematics by the prime minister of India at apublic function in Madras on December 26, 2011.On that day, a stamp of Ramanujan was releasedand Robert Kanigel was given a special awardfor his biography of Ramanujan. Also, the TataInstitute released a reprinting of the Notebooksof Ramanujan in the form of a collector’s edition[47]. The mathematics year has featured severalconferences and public lectures on Ramanujanthroughout India.

The Ramanujan Journal is bringing out a spe-cial volume for the 125th birth anniversary inDecember 2012. The 125th anniversary celebra-tions will conclude in 2012 with an InternationalConference on the Works of Ramanujan at theUniversity of Mysore, India, December 12–13; anInternational Conference on the Legacy of Ra-manujan at SASTRA University in Kumbakonamduring December 14–16; and an International Con-ference on Ramanujan’s Legacy in India’s capitalNew Delhi during December 17–22. In 2012 only,the SASTRA Ramanujan Prize will be given out-side of Kumbakonam, namely, at the New Delhiconference.

Just as there was a Ramanujan CentennialConference at the University of Illinois, Urbana,in the summer of 1987 before the centennialcelebrations in India that December, there was aRamanujan 125 Conference at the University ofFlorida, Gainesville, in November 2012, one monthbefore the 125th anniversary celebrations in India.Finally, at the Joint Annual Meetings of the AMSand MAA in January 2013 in San Diego, there willbe a Special Session on Ramanujan’s mathematicsto mark his 125th birth anniversary.

We have much to be proud of regarding whathas happened in the quarter century after theRamanujan Centennial. The articles in this Noticesfeature convince us that the world of Ramanujan’smathematics is continuing to grow, and so wehave much to look forward to in the next quartercentury.

References[1] S. Ahlgren and K. Ono, Congruence properties for

the partition function, Proc. Nat. Acad. Sci. USA 98(2001), 12882–12884.

[2] K. Alladi, The discovery and rediscovery of mathe-matical genius, Book Review of The Man Who KnewInfinity (by Robert Kanigel), The American Scientist 80(1992), 388–389.

[3] , Book Review of Ramanujan: Letters and Com-mentary (by Bruce C. Berndt and Robert A. Rankin),The American Math. Monthly 103 (1996), 708–713.

[4] , Refinements of Rogers-Ramanujan type iden-tities, in Proc. Fields Inst. Conf. on Special Functions,q-Series and Related Topics, Fields Inst. Communica-tions, 14, Amer. Math. Soc., Providence, RI, 1997, pp.1–35.

[5] , Book Review of Ramanujan: Essays and Sur-veys (by Bruce C. Berndt and Robert A. Rankin),American Math. Monthly 110 (2003), 861–865.

[6] , A pilgrimage to Ramanujan’s hometown, MAAFocus 26 (2006), 4–6.

[7] , The first SASTRA Ramanujan prizes, MAAFocus 26 (2006), 7.

[8] , Ramanujan’s Place in the World of Mathemat-ics, Springer, New Delhi, 2012, 180 pp.

[9] K. Alladi, G. E. Andrews, and B. Gordon, General-izations and refinements of a partition theorem ofGöllnitz, J. Reine Angew. Math. 460 (1995), 165–188.

[10] K. Alladi, G. E. Andrews, and A. Berkovich, Anew four parameter q-series identity and its partitionimplications, Invent. Math. 153 (2003), 231–260.

[11] G. E. Andrews, The theory of partitions, in Encyl. Math.,2, Addison-Wesley, Reading, MA 1976.

[12] , An introduction to Ramanujan’s lost notebook,Amer. Math. Monthly 86 (1979), 89–108.

[13] , q-Series: Their Development and Applicationsin Analysis, Number Theory, Combinatorics, Physics,and Computer Algebra, CBMS Regional Conf. Ser. inMath., 66, Amer. Math. Soc., Providence, RI, 1986.

[14] , The hard-hexagon model and the Rogers-Ramanujan identities, Proc. Nat. Acad. Sci. USA 78(1981), 5290–5292.

[15] , On a partition function of Richard Stanley,Stanley Festschrift, Elec. J. Comb. 11 (2004/06).

[16] , Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent. Math. 169 (2007),37–73.

[17] , The meaning of Ramanujan, now and for thefuture, Ramanujan J. 20 (2009), 257–273.

[18] G. E. Andrews, R. J. Baxter, and P. J. Forrester, Eight-vertex SOS model and generalized Rogers-Ramanujantype identities, J. Stat. Phys. 35 (1984), 193–266.

[19] G. E. Andrews and B. C. Berndt, Ramanujan’s LostNotebook, Parts I–V, Springer, New York, 2005, 2009,2012, to appear.

[20] G. E. Andrews and F. Garvan, Dyson’s crank of apartition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 167–171.

[21] , Ramanujan’s “lost” notebook, VI: The mocktheta conjectures, Adv. in Math. 73 (1989), 242–255.

[22] A. O. L. Atkin, Multiplicative congruence propertiesand density problems for p(n), Proc. London Math.Soc. 18 (1968), 563–576.

December 2012 Notices of the AMS 1527

[23] A. O. L. Atkin and F. Garvan, Relations between theranks and cranks of partitions, Ramanujan J. 7 (2003),343–366.

[24] A. Berkovich and F. Garvan, On the Andrews-Stanleyrefinement of Ramanujan’s partition congruence mod-ulo 5 and generalizations, Trans. Amer. Math. Soc. 358(2006), 703–726.

[25] A. Berkovich and B. McCoy, Continued fractions andFermionic representations for characters of M(p,p′)minimal models, Lett. Math. Phys. 37 (1996), 49–66.

[26] , Rogers-Ramanujan identities: A century ofprogress from mathematics to physics, Doc. Math. 1998Extra Vol. III ICM, 163–172.

[27] A. Berkovich, B. McCoy, A. Schilling, and O. War-naar, Bailey flows and Bose-Fermi identities for con-

formal coset models (A(1)1 )N × (A(1)1 )N′/(A(1)1 )N+N′ ,

Nucl. Phys. B 499 (1997), 621–649.[28] A. Berkovich, B. McCoy, and A. Schilling, Rogers-

Schur-Ramanujan type identities for M(p,p′) modelsof conformal field theory, Comm. Math. Phys. 191(1998), 325–395.

[29] Bruce C. Berndt, Ramanujan’s Notebooks, Parts I–V,Springer, New York, 1985, 1989, 1991, 1994, 1998.

[30] Bruce C. Berndt and Robert A. Rankin, Ramanujan:Letters and Commentary, Amer. Math. Soc., Providence,RI, and LMS, London, 1995.

[31] , Ramanujan: Essays and Surveys, Amer. Math.Soc., Providence, RI, and LMS, London, 2001.

[32] K. Bringmann and K. Ono, The f (q) mock theta func-tion conjecture and partition ranks, Invent. Math. 165(2006), 243–266.

[33] , Dyson’s rank and Maass forms, Annals of Math.171 (2010), 419–449.

[34] J. K. Bruinier and K. Ono, Heegner divisors, L-functions, and harmonic weak Maass forms, Annalsof Math. 172 (2010), 2135–2181.

[35] F. J. Dyson, Some guesses in the theory of partitions,Eureka (Cambridge) 8 (1944), 10–15.

[36] Francis Gerard Garvan, Generalizations of Dyson’srank, Ph.D. Thesis, The Pennsylvania State University(1986), 136 pp.

[37] Frank Garvan, Dongsu Kim, and Dennis Stanton,Cranks and t-cores, Invent. Math. 101 (1990), 1–17.

[38] H. Halberstam, Review of The Indian Clerk (by DavidLeavitt), Notices Amer. Math. Soc. 55 (2008), 952–956.

[39] G. H. Hardy, Ramanujan—Twelve Lectures on his Lifeand Work, Cambridge University Press, Cambridge,1940.

[40] D. Hickerson, A proof of the mock theta conjectures,Invent. Math. 94 (1988), 639–660.

[41] Robert Kanigel, The Man Who Knew Infinity: A Life ofthe Genius Ramanujan, Charles Scribner’s, New York,1991.

[42] David Leavitt, The Indian Clerk, Bloomsbury, NewYork, 2007.

[43] Karl Mahlburg, Partition congruences and theAndrews-Garvan-Dyson crank, Proc. Nat. Acad. Sci.USA 102 (2005), 15373–15376.

[44] K. Ono, Distribution of the partition function modulom, Annals of Math. 151 (2000), 293–307.

[45] , Honoring a gift from Kumbakonam, NoticesAmer. Math. Soc. 53 (2006), 640–651.

[46] S. Ramanujan, Collected Papers, Cambridge UniversityPress, Cambridge, 1927.

[47] , Notebooks (2 volumes), Tata Institute ofFundamental Research, Bombay, 1957 (reprinted asCollector’s Edition in 2011).

[48] , The Lost Notebook and Other UnpublishedPapers, Narosa, New Delhi, 1987.

[49] R. Stanley, Problem 10969, Amer. Math. Monthly 109(2000), 760.

George E. Andrews

Partitions and the Rogers-RamanujanIdentitiesIn his first letter to G. H. Hardy [19, p. xxvii],Ramanujan asserts:

“The coefficient of xn in1

1− 2x+ 2x4 − 2x9 + 2x16 − · · ·

= the nearest integer to1

4n

{cosh(π

√n)− sinh(π

√n)

π√n

}, ”

(1)

and, in a footnote, Hardy states, “This is quiteuntrue. But the formula is extremely interestingfor a variety of reasons.” Perhaps one of the mostimportant justifications for calling it “extremelyinteresting” is that it eventually led to the Hardy-Ramanujan formula for p(n).

Each positive integer can be partitioned intounordered sums of positive integers. For exam-ple, there are seven partitions of the number 5:1+1+1+1+1, 1+1+1+2, 1+2+2, 1+1+3, 2+3, 1+4, 5.We denote by p(n) the number of partitions on n,so p(5) = 7.

Inspired by (1), Hardy and Ramanujan eventuallyproved [13] the following:

Theorem 1. Suppose that

φq(n) =√q

2π√

2

ddn

(eCλn/q/λn

),

where C and λn are defined by C = 2π√6

and λn =√n− 1

24 , and for all positive integral values of q;that p is a positive integer less than and prime toq; that ωp,q is a 24q-th root of unity, defined whenp is odd by the formula

ωp,q =(−qp

)exp

[−{

14(2− pq − p)+ 1

12

(q − 1

q

)

(2p − p′ + p2p′)}πi]

and when q is odd by the formula

ωp,q =(−pq

)exp

[−{

14(q − 1)+ 1

12

(q − 1

q

)

(2p − p′ + p2p′)}πi],

George E. Andrews is Evan Pugh Professor of Mathematicsat The Pennsylvania State University. His email address isandrews@math.psu.edu.

1528 Notices of the AMS Volume 59, Number 11

Ph

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Krishnaswami Alladi and George Andrews infront of Srinivasa Ramanujan’s home,Sarangapani Sannidhi Street, Kumbakonam,December 20, 2003.

where (a/b) is the symbol of Legendre and Jacobiand p′ is any positive integer such that 1+ pp′ isdivisible by q; that

Aq(n) =∑(p)ωp,qe−2npπi/q ;

and that α is any positive constant; and ν, theinteger part of α

√n.

Then

p(n) =ν∑1

Aqφq +O(n−14 ),

so that p(n) is, for all sufficiently large values of n,the integer nearest to

ν∑1

Aqφq.

In 1937 Rademacher was preparing lecturenotes on this amazing theorem. To simplify thingsa bit, he replaced 1

2eCλn/q by sinh(Cλn/q) in the

definition of φq(n) [17, p. 699]. To his amazement,he was now able to prove that, in fact,

p(n) =∞∑q=1

Aqφq.

In the celebrations surrounding the one hun-dredth anniversary of Ramanujan’s birth, AtleSelberg [22] notes a further major surprise:

If one looks at Ramanujan’s first letter toHardy, there is a statement there [ed: i.e.,(1) above] which has some relation to hislater work on the partition function, namelyabout the coefficient of the reciprocal ofa certain theta series (a power series withsquare exponents and alternating signs ascoefficients). It gives the leading term in whathe claims as an approximate expression forthe coefficient. If one looks at that expression,one sees that this is the exact analogue of

the leading term in the Rademacher formulafor p(n) which shows that Ramanujan, inwhatever way he had obtained this, had beenled to the correct term of that expression.

In the work on the partition function,studying the paper, it seems clear to methat it must have been, in a way, Hardy whodid not fully trust Ramanujan’s insight andintuition, when he chose the other form ofthe terms in their expression, for a purelytechnical reason, which one analyses as notvery relevant. I think that if Hardy had trustedRamanujan more, they should have inevitablyended with the Rademacher series. There islittle doubt about that.

The subsequent impact of the “circle method”introduced by Hardy and Ramanujan (cf. [25]) isdiscussed by Vaughan. Further amazing formulasfor p(n) have been found by Folsom, Kent, andOno [10], along with spectacular results on thedivisibility of p(n); these aspects will be discussedby Ono.

The Rogers-Ramanujan identities also form anintriguing and surprising component of Ramanu-jan’s career. In 1916 P. A. MacMahon began ChapterIII, Section VII, of his monumental CombinatoryAnalysis [16, p. 33] as follows:

RAMANUJAN IDENTITIESMr Ramanujan of Trinity College, Cam-

bridge, has suggested a large number offormulae which have applications to thepartition of numbers. Two of the most in-teresting of these concern partitions whoseparts have a definite relation to the modulusfive. Theorem I gives the relation

1+ x1− x +

x4

(1− x)(1− x2)

+ x9

(1− x)(1− x2)(1− x3)+ · · ·

+ xi2

(1− x)(1− x2) . . . (1− xi) + · · ·

= 1(1− x)(1− x6)(1− x11) . . . (1− x5m+1) . . .

× 1(1− x4)(1− x9)(1− x14) . . . (1− x5m+4) . . .

,

where on the right-hand side the exponents ofx are the numbers given by the congruences≡ 1 mod 5,≡ 4 mod 5.

This most remarkable theorem has beenverified as far as the coefficient of x89 byactual expansion so that there is practicallyno reason to doubt its truth; but it has notyet been established.

MacMahon then notes that this identity implies thefollowing assertion about partitions:

December 2012 Notices of the AMS 1529

The theorem asserts that the partitions,whose parts are limited to be of the forms5m + 1,5m + 4, are equi-numerous withthose which involve neither repetitions norsequences.

Subsequently, MacMahon notes that, in the secondidentity,

1+ x2

1− x+x6

(1− x)(1− x2)+ x12

(1− x)(1− x2)(1− x3)+· · ·

+ xi2+i

(1− x)(1− x2) . . . (1− xi) + · · ·

= 1(1− x2)(1− x7)(1− x12) . . . (1− x5m+2)

× 1(1− x3)(1− x8)(1− x13) . . . (1− x5m+3) . . .

,

the exponents of x on the right-hand side are ofthe form 5m+ 2 and 5m+ 3.

This relation has also been verified by actualexpansion to a high power of x, but it has not beenestablished. As with the first identity, MacMahonnotes the implications for partitions:

states that of any given number such parti-tions are equi-numerous with those whoseparts are all of the forms 5m+ 2,5m+ 3.So here in 1916 was an immensely appealing

unsolved problem. As Hardy remarks [14, p. 91]:

Ramanujan rediscovered the formulae some-time before 1913. He had then no proof (andknew that he had none), and none of themathematicians to whom I communicated theformulae could find one. They are thereforestated without proof in the second volumeof MacMahon’s Combinatory Analysis.

The mystery was solved, trebly, in 1917.In that year Ramanujan, looking throughold volumes of the Proceedings of the Lon-don Mathematical Society, came accidentallyacross Rogers’ paper. I can remember verywell his surprise, and the admiration whichhe expressed for Rogers’ work. A correspon-dence followed in the course of which Rogerswas led to a considerable simplification of hisoriginal proof. About the same time I. Schur,who was then cut off from England by thewar, rediscovered the identities again.

In his account of Ramanujan’s notebooks, Berndt[9, pp. 77–79] gives a detailed history of subsequentwork on the Rogers-Ramanujan identities.

Surprisingly, this aspect of the theory of par-titions languished for the next forty years. W. N.Bailey [8], [7] and his student Lucy Slater [24], [23]did generalize the series-product identities, butD. H. Lehmer [15] and H. L. Alder [1] publishedpapers suggesting that there were no generaliza-tions of an obvious sort. Even Hans Rademacher[18, p. 73] asserted that “It can be shown there can

be no corresponding identities for moduli higherthan 5.”

It turned out that Lehmer and Alder hadshown only that the wrong generalization is thewrong generalization. The subject truly opened upwith B. Gordon’s magnificent partition-theoreticgeneralization [12] in 1961.

Theorem 2. Let Bk,i(n) denote the number of par-titions of n of the form b1 + b2 + · · · + bs , wherebj ≥ bj+1, bj − bj+k−1 ≥ 2 and at most i − 1 of thebj = 1. Let Ak,i(n) denote the number of partitionsof n into parts 6≡ 0,±i(mod 2k+ 1). Then Ak,i(n) =Bk,i(n), for 1 ≤ i ≤ k.

The cases k = 2, i = 1,2 are the partitiontheoretic consequences of the original identitiesas stated by MacMahon.

The question of analogous identities remainedopen until 1974, when the following result appeared[15].

Theorem 3. For 1 ≤ i ≤ k,∑n1,...,nk−1≥0

qN21+···+N2

k−1+Ni+Ni+1+···+Nk−1

(q)n1(q)n2 · · · (q)nk−1

=∞∏n=1

n 6≡0,±i(mod 2k+1)

11− qn ,

where Nj = nj + nj+1 + · · · + nk−1, and (q)n =(1− q)(1− q2) · · · (1− qn).

Since the 1970s there has been a cornucopiaof Rogers–Ramanujan-type results pouring forth.As mentioned earlier, Berndt [9, p. 77] gives manyreferences to recent work.

It should be added that Schur (mentioned inHardy’s brief history given above) had a strong in-fluence on the combinatorial/arithmetical aspectsof Rogers-Ramanujan theory. Both his indepen-dent discovery of the Rogers-Ramanujan identities[20] and his modulus 6 theorem of 1926 [21]greatly impacted subsequent developments. Theseinclude the Ph.D. thesis of H. Göllnitz [11] and the“weighted words” method of Alladi and Gordon[2], the latter culminating in a generalization ofGöllnitz’s deep modulus 12 theorem by Alladi,Andrews, and Berkovich [3].

The analytic side of the study was greatlyassisted by the discovery of “Bailey chains” in1984 [6]. Further accounts of this part of theRogers-Ramanujan identities are provided in S. O.Warnaar’s survey [26] “50 Years of Bailey’s lemma”and in Chapter 3 of the Selected Works of George E.Andrews [4].

The Rogers-Ramanujan identities also leaddirectly to a study of a related continued fraction,a topic dealt with at lenghth in [9].

1530 Notices of the AMS Volume 59, Number 11

References[1] Henry L. Alder, The nonexistence of certain identities

in the theory of partitions and compositions, Bull.Amer. Math. Soc., 54:712–722, 1948.

[2] Krishnaswami Alladi, The method of weightedwords and applications to partitions. In Number The-ory (Paris, 1992–1993), volume 215 of London Math.Soc. Lecture Note Ser., pages 1–36, Cambridge Univ.Press, Cambridge, 1995.

[3] Krishnaswami Alladi, George E. Andrews, andAlexander Berkovich, A new four parameter q-series identity and its partition implications, Invent.Math., 153(2):231–260, 2003.

[4] G. E. Andrews and A. V. Sills, Selected Worksof George E. Andrews: With Commentary, ImperialCollege Press, 2012.

[5] George E. Andrews, An analytic generalization ofthe Rogers-Ramanujan identities for odd moduli, Proc.Nat. Acad. Sci. U.S.A., 71:4082–4085, 1974.

[6] , Multiple series Rogers-Ramanujan typeidentities, Pacific J. Math., 114(2):267–283, 1984.

[7] W. N. Bailey, Some identities in combinatory analysis,Proc. London Math. Soc. (2), 49:421–425, 1947.

[8] , Identities of the Rogers-Ramanujan type, Proc.London Math. Soc. (2), 50:1–10, 1948.

[9] Bruce C. Berndt, Ramanujan’s Notebooks. Part III,Springer-Verlag, New York, 1991.

[10] Amanda Folsom, Zachary A. Kent, and Ken Ono,l-adic properties of the partition function, Advancesin Mathematics, 229(3):1586–1609, 2012.

[11] H. Göllnitz, Partitionen mit Differenzenbedingungen,J. Reine Angew. Math., 225:154–190, 1967.

[12] Basil Gordon, A combinatorial generalization of theRogers-Ramanujan identities. Amer. J. Math., 83:393–399, 1961.

[13] G. H. Hardy and S. Ramanujan, Asymptotic formulæin combinatory analysis [Proc. London Math. Soc. (2)17 (1918), 75–115], in Collected Papers of SrinivasaRamanujan, pages 276–309, AMS Chelsea Publ., Amer.Math. Soc., Providence, RI, 2000.

[14] G. H. Hardy, Ramanujan: Twelve Lectures on Sub-jects Suggested by His Life and Work, AMS ChelseaPublishing Series, Amer. Math. Soc., Providence, RI,1999.

[15] D. H. Lehmer, Two nonexistence theorems onpartitions, Bull. Amer. Math. Soc., 52:538–544, 1946.

[16] Percy A. MacMahon, Combinatory Analysis. Vols.I, II (bound in one volume), Dover Phoenix Editions,Dover Publications Inc., Mineola, NY, 2004. Reprint ofAn Introduction to Combinatory Analysis (1920) andCombinatory Analysis. Vols. I, II (1915, 1916).

[17] H. Rademacher, On the partition function p(n), Proc.London Math. Soc., 43(2):241–254, 1937.

[18] , Lectures on Elementary Number Theory, TataInstitute of Fundamental Research, Bombay. Lec-tures on mathematics and physics. Mathematics. TataInstitute of Fundamental Research, 1963.

[19] Srinivasa Ramanujan, Collected Papers of SrinivasaRamanujan, AMS Chelsea Publishing, Amer. Math. Soc.,Providence, RI, 2000, edited by G. H. Hardy, P. V. SeshuAiyar, and B. M. Wilson, third printing of the 1927original, with a new preface and commentary by BruceC. Berndt.

[20] I. Schur, Ein Beitrag zur additiven Zahlentheorie undzur Theorie der Kettenbrüche, S.-B. Preuss. Akad. Wiss.

Ph

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adi.

The only cot at the humble home of Ramanujanin Kumbakonam. As a young boy, Ramanujanused to sit on the windowsill and do his “sums”,watching the passersby on the street.

Phys.-Math. Kl., pages 302–321, 1917. Reprinted inGesammelte Abhandlungen. Springer-Verlag, 1973.

[21] , Zur additiven Zahlentheorie, S.-B. Preuss. Akad.Wiss. Phys.-Math. Kl., pages 488–495, 1926. Reprintedin Gesammelte Abhandlungen, Springer-Verlag, 1973.

[22] A. Selberg, Collected Papers, Vol. I, Springer-Verlag,1989.

[23] L. J. Slater, A new proof of Rogers’s transformationsof infinite series, Proc. London Math. Soc. (2), 53:460–475, 1951.

[24] , Further identities of the Rogers-Ramanujantype, Proc. London Math. Soc. (2), 54:147–167, 1952.

[25] R. C. Vaughan, The Hardy-Littlewood Method, Cam-bridge Tracts in Mathematics, vol. 80, CambridgeUniversity Press, Cambridge, 1981.

[26] S. Ole Warnaar, 50 years of Bailey’s lemma, in Alge-braic Combinatorics and Applications (Gößweinstein,1999), pages 333–347, Springer, Berlin, 2001.

Bruce C. Berndt

Ramanujan’s Earlier Notebooks and His LostNotebook

Since the centenary of Ramanujan’s birth, a largeamount of Ramanujan’s mathematics that hadbeen ensconced in his earlier notebooks andin his lost notebook has been brought to light.In particular, in 1987 only one book had beenwritten on Ramanujan’s notebooks [2], but nowseveral have appeared [2], [1]. We now have alarger portion of Ramanujan’s mathematics toappreciate, evaluate, analyze, and challenge. Thechallenge is to ascertain Ramanujan’s thinking andmotivation and to properly place his work withincontemporary mathematics. Since other writers inthis series of articles are focusing on Ramanujan’s

Bruce C. Berndt is professor of mathematics at the Univer-sity of Illinois, Urbana-Champaign. His email address isberndt@illinois.edu.

December 2012 Notices of the AMS 1531

lost notebook, we give more concentration in thispaper on his earlier notebooks.

For most of his life, both in Kumbakonam andlater in Madras, Ramanujan worked on a slate.Paper was expensive for him, and so he recordedonly his final results in notebooks. Writing downhis proofs in notebooks would have taken precioustime; Ramanujan was clearly anxious to get onwith his next ideas. Moreover, if someone hadasked him how to prove a certain claim in hisnotebooks, he undoubtedly was confident thathe could reproduce his argument. We do notknow precisely when Ramanujan began to recordhis mathematical discoveries in notebooks, butit was likely around the time that he enteredthe Government College in Kumbakonam in 1904.Except for possibly a few entries, he evidentlystopped recording his theorems in notebooks afterhe boarded a passenger ship for England on March17, 1914. That Ramanujan no longer concentratedon logging entries in his notebooks is evident fromtwo letters that he wrote to friends in Madrasduring his first year in England [5, pp. 112–113;123–125]. In a letter of November 13, 1914, to hisfriend R. Krishna Rao, Ramanujan confided, “I havechanged my plan of publishing my results. I amnot going to publish any of the old results in mynotebooks till the war is over.” And in a letter ofJanuary 7, 1915, to S. M. Subramanian, Ramanujanadmitted, “I am doing my work very slowly. Mynotebook is sleeping in a corner for these fouror five months. I am publishing only my presentresearches as I have not yet proved the results inmy notebooks rigorously.”

Prior to his journey to England in 1914, Ra-manujan prepared an enlarged edition of his firstnotebook; this augmented edition is the secondnotebook. A third notebook contains only thirty-three pages. G. N. Watson, in his delightful accountof the notebooks [16], relates that Sir Gilbert Walker,head of the Indian Meteorological Observatory anda strong supporter of Ramanujan, told him that inApril 1912 he observed that Ramanujan possessedfour or five notebooks, each with a black cover andeach about an inch thick. We surmise that thesenotebooks were either a forerunner of the firstnotebook or that the first notebook is an amalgamof these earlier black-covered notebooks.

When Ramanujan returned to India on March 13,1919, he took with him only his second andthird notebooks. The first notebook was left withG. H. Hardy, who used it to write a paper [9] on achapter in the notebook devoted to hypergeometricseries. Hardy’s paper [9] was read on February 5,1923, and published later during that year. Hebegins by writing, “I have in my possession amanuscript note-book which Ramanujan left withme when he returned to India in 1919.” It was

not until March 1925 that Hardy returned thefirst notebook to the University of Madras throughits librarian, S. R. Rangathanan [15, pp. 56, 57]. Ahandwritten copy was subsequently made and sentto Hardy in August 1925.

Meanwhile, Hardy had originally wanted to bringRamanujan’s published papers, his notebooks, andany other unpublished manuscripts together forpublication. To that end, in a letter to Hardy datedAugust 30, 1923, Francis Dewsbury, registrar atthe University of Madras, wrote, “I have the honourto advise despatch to-day to your address perregistered and insured parcel post of the fourmanuscript note-books referred to in my letterNo. 6796 of the 2nd idem. I also forward a packet ofmiscellaneous papers which have not been copied.It is left to you to decide whether any or all ofthem should find a place in the proposed memorialvolume. Kindly preserve them for ultimate returnto this office.” The “four manuscript notebooks”were handwritten copies of the second and thirdnotebooks, and the packet of miscellaneous papersevidently included the “lost notebook”, whichHardy passed on to Watson and which was laterrediscovered by George Andrews in the TrinityCollege Library in March 1976. Note that, incontradistinction to Dewsbury’s request, Hardynever returned the miscellaneous papers.

As it transpired, only Ramanujan’s publishedpapers were collected for publication. However,published with the Collected Papers [12] werethe first two letters that Ramanujan had writtento Hardy, which contained approximately onehundred twenty mathematical claims. Upon theirpublication, these letters generated considerableinterest, with the further publication of severalpapers establishing proofs of these claims. Con-sequently, either in 1928 or 1929, at the strongsuggestion of Hardy, Watson and B. M. Wilson,one of the three editors of [12], agreed to edit thenotebooks. The word “edit” should be interpretedin a broader sense than usual. If a proof of a claimcan be found in the literature, then it is sufficientto simply cite sources where proofs can be found;if a proof cannot be found in the literature, thenthe task of the editors is to provide one. In anaddress [16] to the London Mathematical Societyon February 5, 1931, Watson cautioned (in retro-spect, far too optimistically), “We anticipate thatit, together with the kindred task of investigatingthe work of other writers to ascertain which of hisresults had been discovered previously, may takeus five years.” Wilson died prematurely in 1935,and although Watson wrote approximately thirtypapers on Ramanujan’s work, his interest evidentlyflagged in the late 1930s, and so the editing wasnot completed.

1532 Notices of the AMS Volume 59, Number 11

Ph

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The majestic gopuram (entrance tower) of theSarangapani Temple as seen from the front ofRamanujan’s home on Sarangapani SannidhiStreet.

Altogether, the notebooks contain over threethousand claims, almost all without proof. Hardysurmised that over two-thirds of these results wererediscoveries. This estimate is much too high; onthe contrary, at least two-thirds of Ramanujan’sclaims were new at the time that he wrote them,and two-thirds more likely should be replacedby a larger fraction. Almost all of the resultsare correct; perhaps no more than five to tenclaims are incorrect. The topics examined byRamanujan in his notebooks fall primarily underthe purview of analysis, number theory, and ellipticfunctions, with much of his work in analysis beingassociated with number theory and with someof his discoveries also having connections withenumerative combinatorics and modular forms.Chapter 16 in the second notebook represents aturning point, since in this chapter he begins toexamine q-series for the first time and also to beginan enormous devotion to theta functions. Chapters16–21 and considerable material in the unorganizedpages that follow focus on Ramanujan’s distinctivetheory of elliptic functions, in particular, on thetafunctions. Especially in the years 1912–1914 and1917–1920, Ramanujan’s concentration on numbertheory was through q-series and elliptic functions.

Finally, in 1957, the notebooks were madeavailable to the public when the Tata Instituteof Fundamental Research in Bombay published aphotocopy edition [13], but no editing was under-taken. The notebooks were set in two volumes,with the first containing the first notebook and

with the second comprising the second and thirdnotebooks. The reproduction is quite faithful, witheven Ramanujan’s scratch work photographed.Buttressed by vastly improved technology, a new,much clearer, edition was published in 2011 to com-memorate the 125th anniversary of Ramanujan’sbirth.

In February 1974, while reading two papers byEmil Grosswald [7], [8] in which some formulasfrom the notebooks were proved, we observedthat we could prove these formulas by usinga transformation formula for a general classof Eisenstein series that we had proved twoyears earlier. We found a few more formulas inthe notebooks that could be proved using ourmethods, but a few thousand further assertionsthat we could not prove. In May 1977 the authorbegan to devote all of his attention to provingall of Ramanujan’s claims in the notebooks. Withthe help of a copy of the notes from Watson andWilson’s earlier attempt at editing the notebooksand with the help of several other mathematicians,the task was completed in five volumes [2] inslightly over twenty years.

In this series of articles for the Notices, GeorgeAndrews, Ken Ono, and Ole Warnaar, in particular,write about Ramanujan’s discoveries in his lostnotebook, which is dominated by q-series. However,published with the lost notebook [14] are severalpartial manuscripts and fragments of manuscriptson other topics that provide valuable insightinto Ramanujan’s interests and ideas. We firstmention two topics that we have examined withSun Kim and Alexandru Zaharescu in two seriesof papers, with [3] and [4] being examples ofthe two series. First, Ramanujan had a stronginterest in the circle problem and Dirichlet divisorproblem, with two deep formulas in the lostnotebook clearly aimed at attacking these problems.Second, [14] contains three very rough, partialmanuscripts on diophantine approximation. In oneof these, Ramanujan derives the best diophantineapproximation to e2/a, where a is a nonzero integer,a result not established in the literature until 1978[6]. Also appearing in [14] are fragments ofmaterial that were originally intended for portionsof published papers, for example, [10] and [11]. Itis then natural to ask why Ramanujan chose not toinclude these results in his papers. In at least twoinstances, the discarded material depends on theuse of complex analysis, a subject that Ramanujanoccasionally employed but not always correctly.In particular, in the cases at hand, Ramanujandid not have a firm knowledge of the Mittag-Leffler Theorem for developing partial fractiondecompositions. The fragments and incompletemanuscripts will be examined in the fourth book

December 2012 Notices of the AMS 1533

that the author is writing with Andrews on the lostnotebook [1].

We conclude with a few remarks about Ra-manujan’s methods. It has been suggested thathe discovered his results by “intuition” or bymaking deductions from numerical calculationsor by inspiration from Goddess Namagiri. Indeed,like most mathematicians, Ramanujan evidentlymade extensive calculations that provided guid-ance. However, Hardy and this writer firmly believethat Ramanujan created mathematics as any othermathematician would and that his thinking canbe explained like that of other mathematicians.However, because Ramanujan did not leave us anyproofs for the vast number of results found inhis earlier notebooks and in his lost notebook,we often do not know Ramanujan’s reasoning. AsRamanujan himself was aware, some of his argu-ments were not rigorous by then-contemporarystandards. Nonetheless, despite his lack of rigor attimes, Ramanujan doubtless thought and devisedproofs as would any other mathematician.

References[1] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost

Notebook, Parts I–V, Springer, New York, 2005, 2009,2012, to appear.

[2] B. C. Berndt, Ramanujan’s Notebooks, Parts I–V,Springer-Verlag, New York, 1985, 1989, 1991, 1994,1998.

[3] B. C. Berndt, S. Kim, and A. Zaharescu, Circle anddivisor problems, and double series of Bessel functions,Adv. Math, to appear.

[4] , Diophantine approximation of the exponentialfunction, and a conjecture of Sondow, submitted forpublication.

[5] B. C. Berndt and R. A. Rankin, Ramanujan: Lettersand Commentary, American Mathematical Society,Providence, RI, 1995; London Mathematical Society,London, 1995.

[6] C. S. Davis, Rational approximation to e, J. Aus-tral. Math. Soc. 25 (1978), 497–502.

[7] E. Grosswald, Die Werte der RiemannschenZetafunktion an ungeraden Argumentstellen,Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1970(1970), 9–13.

[8] , Comments on some formulae of Ramanujan,Acta Arith. 21 (1972), 25–34.

[9] G. H. Hardy, A chapter from Ramanujan’s note-book,Proc. Cambridge Philos. Soc. 21 (1923), 492–503.

[10] S. Ramanujan, Some formulae in the analytic theoryof numbers, Mess. Math. 45 (1916), 81–84.

[11] , On certain trigonometric sums and their appli-cations in the theory of numbers, Trans. CambridgePhilos. Soc. 22 (1918), 259–276.

[12] , Collected Papers, Cambridge University Press,Cambridge, 1927; reprinted by Chelsea, New York,1962; reprinted by the American Mathematical Society,Providence, RI, 2000.

[13] , Notebooks (2 volumes), Tata Institute of Funda-mental Research, Bombay, 1957; second edition, TataInstitute of Fundamental Research, Mumbai, 2011.

[14] , The Lost Notebook and Other UnpublishedPapers, Narosa, New Delhi, 1988.

[15] S. R. Ranganathan, Ramanujan: The Man and theMathematician, Asia Publishing House, Bombay, 1967.

[16] G. N. Watson, Ramanujan’s notebooks, J. LondonMath. Soc. 6 (1931), 137–153.

Jonathan M. Borwein

Ramanujan and PiSince Ramanujan’s 1987 centennial, much newmathematics has been stimulated by uncannyformulas in Ramanujan’s Notebooks (lost andfound). In illustration, I mention the expositionby Moll and his colleagues [1] which illustratesvarious neat applications of Ramanujan’s MasterTheorem, which extrapolates the Taylor coefficientsof a function, and relates them to methods ofintegration used in particle physics. I also notelovely work on the modular functions behindApéry and Domb numbers by Chan and others [6],and finally I mention my own work with Crandallon Ramanujan’s arithmetic-geometric continuedfraction [12].

For reasons of space, I now discuss only workrelated directly to pi, and so continue a story startedin [9], [11]. Truly novel series for 1/π , based onelliptic integrals, were found by Ramanujan around1910 [19], [5], [7], [21]. One is

(1)1π= 2

√2

9801

∞∑k=0

(4k)! (1103+ 26390k)(k!)43964k .

Each term of (1) adds eight correct digits. Thoughthen unproven, Gosper used (1) for the computationof a then-record 17 million digits of π in 1985,thereby completing the first proof of (1) [7, Ch. 3].Soon after, David and Gregory Chudnovsky foundthe following variant, which relies on the quadraticnumber field Q(

√−163) rather than Q(

√58), as is

implicit in (1):

(2)1π= 12

∞∑k=0

(−1)k (6k)! (13591409+ 545140134k)(3k)! (k!)3 6403203k+3/2 .

Each term of (2) adds fourteen correct digits. (Werea larger imaginary quadratic field to exist withclass number one, there would be an even moreextravagant rational series for some surd dividedby π [10].) The brothers used this formula severaltimes, culminating in a 1994 calculation of π toover four billion decimal digits. Their remarkablestory was told in a prize-winning New Yorker article[18]. Remarkably, (2) was used again in 2010 and2011 for the current record computations of π tofive and ten trillion decimal digits respectively.

Jonathan M. Borwein is Laureate Professor in the Schoolof Mathematical and Physical Sciences at the Univer-sity of Newcastle (NSW), Australia. His email address isjon.borwein@gmail.com.

1534 Notices of the AMS Volume 59, Number 11

Quartic Algorithm for πThe record for computation of π has gone from29.37 million decimal digits in 1986 to ten trilliondigits in 2011. Since the algorithm below, whichfound its inspiration in Ramanujan’s 1914 paper,was used as part of computations both then andas late as 2009, it is interesting to compare theperformance in each case: Set a0 := 6− 4

√2 and

y0 :=√

2− 1; then iterate

yk+1=1− (1− y4

k )1/4

1+ (1− y4k )1/4

,

ak+1=ak(1+ yk+1)4−22k+3yk+1(1+yk+1+y2k+1).

(3)

Thenak converges quartically to 1/π ; each iterationquadruples the number of correct digits. Twenty-one iterations produce an algebraic number thatcoincides with π to well over six trillion places.

This scheme and the 1976 Salamin–Brentscheme [7, Ch. 3] have been employed fre-quently over the past quarter century. Hereis a highly abbreviated chronology (based onhttp://en.wikipedia.org/wiki/Chronology_of_computation_of_pi):

• 1986: David Bailey used (3) to compute29.4 million digits of π . This required 28hours on one CPU of the new Cray-2 at NASAAmes Research Center. Confirmation usingthe Salamin-Brent scheme took another40 hours. This computation uncoveredhardware and software errors on the Cray-2.

• January 2009: Takahashi used (3) to com-pute 1.649 trillion digits (nearly 60,000times the 1986 computation), requiring73.5 hours on 1,024 cores (and 6.348 Tbytememory) of the Appro Xtreme-X3 system.Confirmation via the Salamin-Brent schemetook 64.2 hours and 6.732 Tbyte of mainmemory.

• April 2009: Takahashi computed 2.576trillion digits.

• December 2009: Bellard computed nearly2.7 trillion decimal digits (first in binary)using (2). This took 131 days, but he usedonly a single four-core workstation withlots of disk storage and even more humanintelligence!

• August 2010: Kondo and Yee computed5 trillion decimal digits, again usingequation (2). This was done in binary,then converted to decimal. The binarydigits were confirmed by computing 32hexadecimal digits of π ending withposition 4,152,410,118,610 using BBP-typeformulas for π due to Bellard and Plouffe[7, Chapter 3]. Additional details are given

1975 1980 1985 1990 1995 2000 2005 2010

108

1010

1012

Figure 1. Plot of πππ calculations in digits (dots)compared with the long-term slope of Moore’sLaw (line).

athttp://www.numberworld.org/misc_runs/pi-5t/announce_en.html. Seealso [4], in which analysis showing thesedigits appear to be “very normal” is made.

Daniel Shanks, who in 1961 computed π toover 100,000 digits, once told Phil Davis thata billion-digit computation would be “foreverimpossible”. But both Kanada and the Chudnovskysachieved that in 1989. Similarly, the intuitionistsBrouwer and Heyting asserted the “impossibility”of ever knowing whether the sequence 0123456789appears in the decimal expansion of π ; yet it wasfound in 1997 by Kanada, beginning at position17387594880. As late as 1989, Roger Penroseventured, in the first edition of his book TheEmperor’s New Mind, that we likely will neverknow if a string of ten consecutive 7s occursin the decimal expansion of π . This string wasfound in 1997 by Kanada, beginning at position22869046249.

Figure 6 shows the progress of π calculationssince 1970, superimposed with a line that chartsthe long-term trend of Moore’s Law. It is worthnoting that whereas progress in computing πexceeded Moore’s Law in the 1990s, it has lagged abit in the past decade. Most of this progress is stillin mathematical debt to Ramanujan.

As noted, one billion decimal digits were firstcomputed in 1989, and the ten (actually fifty) billiondigit mark was first passed in 1997. Fifteen yearslater one can explore, in real time, multibillion stepwalks on the hex digits of π at http://carmaweb.newcastle.edu.au/piwalk.shtml, as drawn byFran Aragon.

Formulas for 1/π2 and More

About ten years ago Jésus Guillera found variousRamanujan-like identities for 1/πN using integer

December 2012 Notices of the AMS 1535

relation methods. The three most basic—andentirely rational—identities are

4π2=

∞∑n=0

(−1)nr(n)5(13+180n+820n2)(

132

)2n+1,

(4)

2π2=∞∑n=0

(−1)nr(n)5(1+ 8n+ 20n2)(

12

)2n+1,

(5)

4π3

?=∞∑n=0

r(n)7(1+14n+76n2+168n3)(

18

)2n+1

,

(6)

where r(n) := (1/2 · 3/2 · · · · · (2n− 1)/2)/n! .Guillera proved (4) and (5) in tandem by very

ingeniously using the Wilf-Zeilberger algorithm[20], [17] for formally proving hypergeometric-likeidentities [7], [15], [21]. No other proof is known.The third, (6), is almost certainly true. Guilleraascribes (6) to Gourevich, who found it usinginteger relation methods in 2001.

There are other sporadic and unexplained exam-ples based on other symbols, most impressively a2010 discovery by Cullen:(7)

211

π4?=∞∑n=0

( 14 )n(

12 )

7n(

34 )n

(1)9n

×(21+ 466n+ 4340n2 + 20632n3 + 43680n4)(

12

)12n.

We shall revisit this formula below.

Formulae for π2

In 2008 Guillera [15] produced another lovely, ifnumerically inefficient, pair of third-millenniumidentities—discovered with integer relation meth-ods and proved with creative telescoping—thistime for π2 rather than its reciprocal. They arebased on:

(8)∞∑n=0

122n

(x+ 1

2

)3

n(x+ 1)3n

(6(n+ x)+ 1) = 8x∞∑n=0

(12

)2

n(x+ 1)2n

,

and

(9)∞∑n=0

126n

(x+ 1

2

)3

n(x+ 1)3n

(42(n+ x)+ 5) = 32x∞∑n=0

(x+ 1

2

)2

n(2x+ 1)2n

.

Here (a)n = a(a + 1) · · · (a + n − 1) is the risingfactorial. Substituting x = 1/2 in (8) and (9), heobtained respectively the formulae

∞∑n=0

122n

(1)3n(32

)3

n

(3n+ 2) = π2

4,(10)

∞∑n=0

126n

(1)3n(32

)3

n

(21n+ 13) = 4π2

3.

Calabi-Yau Equations and Supercongruences

Motivated by the theory of Calabi-Yau differentialequations [2], Almkvist and Guillera have discov-ered many new identities. One of the most pleasingis

(11)1π2

?= 323

∞∑n=0

(6n)!(n!)6

(532n2 + 126n+ 9

)106n+3

.

This is yet one more case where mysteriousconnections have been found between disparateparts of mathematics and Ramanujan’s work [21],[13], [14].

As a final example, we mention the existence ofsupercongruences of the type described in [3], [16],[23]. These are based on the empirical observationthat a Ramanujan series for 1/πN , if truncatedafter p − 1 terms for a prime p, seems always toproduce congruences to a higher power of p. Theformulas below are taken from [22]:

(12)p−1∑n=0

( 14 )n(

12 )

3n(

34 )n

24n (1)5n

(3+ 34n+ 120n2

)≡ 3p2(modp5),

p−1∑n=0

( 14 )n(

12 )

7n(

34 )n

212n (1)9n

(21+ 466n+ 4340n2 + 20632n3 + 43680n4

)(13)

?≡ 21p4(modp9).

We note that (13) is the supercongruence corre-sponding to (6), while for (12) the correspondinginfinite series sums to 32/π4. We conclude byreminding the reader that all identities marked

with ‘?=’ are assuredly true but remain to be proved.

Ramanujan might well be pleased.

References[1] T. Amdeberhan, O. Espinosa, I. Gonzalez, M. Har-

rison, V. Moll, and A. Straub, Ramanujan’s MasterTheorem, Ramanujan J. 29 (2012, to appear).

[2] G. Almkvist, Ramanujan-like formulas for 1/π2 à laGuillera and Zudilin and Calabi-Yau differential equa-tions, Computer Science Journal of Moldova 17, no. 1(49) (2009), 100–120.

[3] G. Almkvist and A. Meurman, Jesus Guillera’s for-mula for 1/π2 and supercongruences (Swedish),Normat 58, no. 2 (2010), 49–62.

[4] D. H. Bailey, J. M. Borwein, C. S. Calude, M. J.Dinneen, M. Dumitrescu, and A. Yee, An empir-ical approach to the normality of pi, ExperimentalMathematics. Accepted February 2012.

[5] N. D. Baruah, B. C. Berndt, and H. H. Chan, Ramanu-jan’s series for 1/π : A survey, Amer. Math. Monthly116 (2009), 567–587.

[6] B. C. Berndt, H. H. Chan, and S. S. Huang, Incompleteelliptic integrals in Ramanujan’s lost notebook, pp.79–126 in q-series from a Contemporary Perspective,Contemp. Math., 254, Amer. Math. Soc., Providence,RI, 2000.

1536 Notices of the AMS Volume 59, Number 11

[7] J. M. Borwein and D. H. Bailey, Mathematics by Ex-periment: Plausible Reasoning in the 21st Century, 2nded., A K Peters, Natick, MA, 2008.

[8] J. M. Borwein, D. H. Bailey, and R. Girgensohn, Ex-perimentation in Mathematics: Computational Roadsto Discovery, A K Peters, 2004.

[9] J. M. Borwein and P. B. Borwein, Ramanujan andpi, Scientific American, February 1988, 112–117.Reprinted in Ramanujan: Essays and Surveys, B. C.Berndt and R. A. Rankin, eds., AMS-LMS History ofMathematics, vol. 22, 2001, pp. 187–199.

[10] , Class number three Ramanujan type seriesfor 1/π , Journal of Computational and Applied Math.(Special Issue) 46 (1993), 281–290.

[11] J. M. Borwein, P. B. Borwein, and D. A. Bailey, Ra-manujan, modular equations and pi or how to computea billion digits of pi, MAA Monthly 96 (1989), 201–219.Reprinted on http://www.cecm.sfu.ca/organics,1996.

[12] J. M. Borwein and R. E. Crandall, On the RamanujanAGM fraction, Part II: The complex-parameter case,Experimental Mathematics 13 (2004), 287–296.

[13] H. H. Chan, J. Wan, and W. Zudilin, Legendre poly-nomials and Ramanujan-type series for 1/π , Israel J.Math., in press, 2012.

[14] , Complex series for 1/π , Ramanujan J., inpress, 2012.

[15] J. Guillera, Hypergeometric identities for 10 ex-tended Ramanujan-type series, Ramanujan J. 15(2008), 219–234.

[16] J. Guillera and W. Zudilin, “Divergent” Ramanujan-type supercongruences, Proc. Amer. Math. Soc. 140(2012), 765–777.

[17] M. Petkovsek, H. S. Wilf, and D. Zeilberger, A = B,A K Peters, 1996.

[18] R. Preston, The Mountains of Pi, New York-er, March 2, 1992, http://www.newyorker.com/archive/content/articles/050411fr_archive01.

[19] S. Ramanujan, Modular equations and approx-imations to pi, Quart. J. Math. 45 (1913–14),350–372.

[20] H. S. Wilf and D. Zeilberger, Rational functions cer-tify combinatorial identities, Journal of the AMS 3(1990), 147–158.

[21] W. Zudilin, Ramanujan-type formulae for 1/π : a sec-ond wind? Modular forms and string duality, in FieldsInst. Commun., 5, Amer. Math. Soc., Providence, RI,2008, pp. 179–188.

[22] , Arithmetic hypergeometric series, RussianMath. Surveys 66:2 (2011), 1–51.

[23] , Ramanujan-type supercongruences, J. NumberTheory 129 (2009), 1848–1857.

The second installment of this article—with piecesby Ken Ono, K. Soundararajan, R. C. Vaughan, andS. Ole Warnaar—will appear in the January 2013issue of the Notices.

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